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Register a new userPhysical expander in Virtual Tree Overlay
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Added by
Maria Potop-Butucaru
Publication date
2011
fields
Informatics Engineering
and research's language is
English
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In this paper, we propose a new construction of constantdegree expanders motivated by their application in P2P overlay networks and in particular in the design of robust trees overlay. Our key result can be stated as follows. Consider a complete binary tree T and construct a random pairing {Pi} between leaf nodes and internal nodes. We prove that the graph GPi obtained from T by contracting all pairs (leaf-internal nodes) achieves a constant node expansion with high probability. The use of our result in improving the robustness of tree overlays is straightforward. That is, if each physical node participating to the overlay manages a random pair that couples one virtual internal node and one virtual leaf node then the physical-node layer exhibits a constant expansion with high probability. We encompass the difficulty of obtaining this random tree virtualization by proposing a local, selforganizing and churn resilient uniformly-random pairing algorithm with O(log2 n) running time. Our algorithm has the merit to not modify the original tree virtual overlay (we just control the mapping between physical nodes and virtual nodes). Therefore, our scheme is general and can be applied to a large number of tree overlay implementations. We validate its performances in dynamic environments via extensive simulations.
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We present improved distributed algorithms for triangle detection and its variants in the CONGEST model. We show that Triangle Detection, Counting, and Enumeration can be solved in $tilde{O}(n^{1/2})$ rounds. In contrast, the previous state-of-the-art bounds for Triangle Detection and Enumeration were $tilde{O}(n^{2/3})$ and $tilde{O}(n^{3/4})$, respectively, due to Izumi and LeGall (PODC 2017). The main technical novelty in this work is a distributed graph partitioning algorithm. We show that in $tilde{O}(n^{1-delta})$ rounds we can partition the edge set of the network $G=(V,E)$ into three parts $E=E_mcup E_scup E_r$ such that (a) Each connected component induced by $E_m$ has minimum degree $Omega(n^delta)$ and conductance $Omega(1/text{poly} log(n))$. As a consequence the mixing time of a random walk within the component is $O(text{poly} log(n))$. (b) The subgraph induced by $E_s$ has arboricity at most $n^{delta}$. (c) $|E_r| leq |E|/6$. All of our algorithms are based on the following generic framework, which we believe is of interest beyond this work. Roughly, we deal with the set $E_s$ by an algorithm that is efficient for low-arboricity graphs, and deal with the set $E_r$ using recursive calls. For each connected component induced by $E_m$, we are able to simulate congested clique algorithms with small overhead by applying a routing algorithm due to Ghaffari, Kuhn, and Su (PODC 2017) for high conductance graphs.
An $(epsilon,phi)$-expander decomposition of a graph $G=(V,E)$ is a clustering of the vertices $V=V_{1}cupcdotscup V_{x}$ such that (1) each cluster $V_{i}$ induces subgraph with conductance at least $phi$, and (2) the number of inter-cluster edges is at most $epsilon|E|$. In this paper, we give an improved distributed expander decomposition. Specifically, we construct an $(epsilon,phi)$-expander decomposition with $phi=(epsilon/log n)^{2^{O(k)}}$ in $O(n^{2/k}cdottext{poly}(1/phi,log n))$ rounds for any $epsilonin(0,1)$ and positive integer $k$. For example, a $(0.01,1/text{poly}log n)$-expander decomposition can be computed in $O(n^{gamma})$ rounds, for any arbitrarily small constant $gamma>0$. Previously, the algorithm by Chang, Pettie, and Zhang can construct a $(1/6,1/text{poly}log n)$-expander decomposition using $tilde{O}(n^{1-delta})$ rounds for any $delta>0$, with a caveat that the algorithm is allowed to throw away a set of edges into an extra part which forms a subgraph with arboricity at most $n^{delta}$. Our algorithm does not have this caveat. By slightly modifying the distributed algorithm for routing on expanders by Ghaffari, Kuhn and Su [PODC17], we obtain a triangle enumeration algorithm using $tilde{O}(n^{1/3})$ rounds. This matches the lower bound by Izumi and Le Gall [PODC17] and Pandurangan, Robinson and Scquizzato [SPAA18] of $tilde{Omega}(n^{1/3})$ which holds even in the CONGESTED CLIQUE model. This provides the first non-trivial example for a distributed problem that has essentially the same complexity (up to a polylogarithmic factor) in both CONGEST and CONGESTED CLIQUE. The key technique in our proof is the first distributed approximation algorithm for finding a low conductance cut that is as balanced as possible. Previous distributed sparse cut algorithms do not have this nearly most balanced guarantee.
The performance of distributed and data-centric applications often critically depends on the interconnecting network. Applications are hence modeled as virtual networks, also accounting for resource demands on links. At the heart of provisioning such virtual networks lies the NP-hard Virtual Network Embedding Problem (VNEP): how to jointly map the virtual nodes and links onto a physical substrate network at minimum cost while obeying capacities. This paper studies the VNEP in the light of parameterized complexity. We focus on tree topology substrates, a case often encountered in practice and for which the VNEP remains NP-hard. We provide the first fixed-parameter algorithm for the VNEP with running time $O(3^r (s+r^2))$ for requests and substrates of $r$ and $s$ nodes, respectively. In a computational study our algorithm yields running time improvements in excess of 200x compared to state-of-the-art integer programming approaches. This makes it comparable in speed to the well-established ViNE heuristic while providing optimal solutions. We complement our algorithmic study with hardness results for the VNEP and related problems.
In this note, we study the expander decomposition problem in a more general setting where the input graph has positively weighted edges and nonnegative demands on its vertices. We show how to extend the techniques of Chuzhoy et al. (FOCS 2020) to this wider setting, obtaining a deterministic algorithm for the problem in almost-linear time.
The minimum spanning tree (MST) construction is a classical problem in Distributed Computing for creating a globally minimized structure distributedly. Self-stabilization is versatile technique for forward recovery that permits to handle any kind of transient faults in a unified manner. The loop-free property provides interesting safety assurance in dynamic networks where edge-cost changes during operation of the protocol. We present a new self-stabilizing MST protocol that improves on previous known ap- proaches in several ways. First, it makes fewer system hypotheses as the size of the network (or an upper bound on the size) need not be known to the participants. Second, it is loop-free in the sense that it guarantees that a spanning tree structure is always preserved while edge costs change dynamically and the protocol adjusts to a new MST. Finally, time complexity matches the best known results, while space complexity results show that this protocol is the most efficient to date.
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